I know that by Heisenberg's Uncertainty Principle that it is not possible to know the exact values of position and momentum of a particle simultaneously, but can we know the exact values of momentum and velocity of a particle simultaneously? I would think the answer would be no because even if we were 100% certain of the particle's position, we would be completely unsure of the particle's momentum, thus making us also completely unsure of the particle's velocity. Does anyone have any insight into this?

4 Answers
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It is quite common to discuss the two extremes of the uncertainty principle, sinusoid and delta function. One has a perfectly defined wavelength but no position, the other has a perfectly defined position but no wavelength.

However, neither one of these shapes is terribly physical for a particle's position wavefunction. A true sinusoidal wavefunction would extend through all space, which is absurd for several reasons (including the presence of other matter). A true delta function would be equally likely to have any momentum, which would probably violate conservation of energy. So, these two extreme limits are mathematically interesting, but not physically relevant.

Given the question "Does the uncertainty principle put some bound on momentum and velocity being simultaneously well-defined?", the answer is no.

Given the question "Does the uncertainty principle forbid me from measuring any single variable with infinite precision?", the answer is no.

Given the question "Does anything forbid me from measuring with infinite precision?", the answer is yes.

So, your question mentions 'exact values', which is a very interesting, thorny subject. (Is it ever possible to measure an exact value? How would we tell the difference?) Are you really curious about 'exact values'? Are you more curious about where the Heisenberg uncertainty principle does and does not apply? Or are you curious if there are other bounds on our ability to measure, in addition to the uncertainty principle?

I was only asking because it was asked on a test and I was curious to know the answer after I took the test. I know the Uncertainty principle deals with energy and time, and then it also deals with position and momentum. So I thought if we hypothetically measured position with exact certainty, then we would be completely uncertain about its position, thus completely uncertain about its velocity. All I wanted to know was if uncertainty about position ensures uncertainty about velocity
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Greg HarringtonApr 21 '11 at 17:05

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If we ignore relativistic effects, then velocity and momentum are directly proportional to each other with the particle's rest mass as the constant of proportionality, so if you know one exactly, you get the other one for free.
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LagerbaerDec 3 '11 at 16:38

If in your theory the momentum operator and velocity operator are proportional to each other, then yes. Knowing one's eigenvalue means knowing the other's. It is always the case with any function of a "known" operator.

The velocity eigenvalues of the Dirac equation are $\pm c$. This is well known since the equation was found; see Dirac's book, "The Principles of Quantum Mechanics, 4th Ed.,", Oxford University Press, Oxford 1958, Chapter XI "Relativistic Theory of the Electron", Section 69, "The motion of a free electron", page 262. It used to be a commonly taught fact of quantum mechanics, but I understand the down votes, it's now possible to get a PhD in physics without knowing the slightest thing about the following quite elementary calculation. Partly since this isn't taught much anymore, the derivation has been reappearing recently in the literature, for example see: Eur.Phys.J.C50:673-678,2007 Chiral oscillations in terms of the zitterbewegung effect / hep-th/0701091, around equation (11).

We begin by noting that velocity is the time rate of change of position, and that you can define the time rate of change of position by using the commutator:
$$\hat{v}_x = \dot{x} = -(i/\hbar)[\hat{x},H]$$
If the above appears to be magic to you, read the wikipedia entry on Ehrenfest's theorem which states the principle and gives the identical situation for non-relativistic quantum mechanics:
$$\frac{d}{dt}\langle x\rangle = -(i/\hbar)\langle [\hat{x},H]\rangle = \langle p_x\rangle /m$$
and so $\;m v_x = m\dot{x} = p_x$ (for the non-relativistic case). Thus, for the non-relativistic electron model, it is possible to simultaneously measure velocity and momentum; their proportionality constant is the mass. But with relativity the proportionality does not happen so the situation is different.

For a state to be an eigenstate of velocity requires that:
$$\hat{v}_x\;\psi(x) = -(i/\hbar)[\hat{x},H]\;\psi(x) = \lambda\psi(x)$$
Dirac defined the free-particle Hamiltonian as $H=c\vec{\alpha}\cdot \vec{p} + \beta mc^2$. In modern notation, $\beta=\gamma^0$ and $\alpha^k = \gamma^0\gamma^k$, while $p$ is the usual momentum operator.

Using the wikipedia's choice of gamma matrix representation, we have:
$$c\gamma^0\gamma^1 = c\left(\begin{array}{cccc}
1&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&-1\end{array}\right)
\left(\begin{array}{cccc}
0&0&0&1\\0&0&1&0\\0&-1&0&0\\-1&0&0&0\end{array}\right)
=c\left(\begin{array}{cccc}
0&0&0&1\\0&0&1&0\\0&1&0&0\\1&0&0&0\end{array}\right)$$
The eigenvalues are obtained by solving the characteristic polynomial. That is, compute the matrix determinant and set it to zero:
$$\left[\begin{array}{cccc}
-\lambda&0&0&c\\0&-\lambda&c&0\\0&c&-\lambda&0\\c&0&0&-\lambda\end{array}\right] = \lambda^4-2\lambda^2c^2 + c^4=0$$
I leave it as an exercise for the reader to show that there are two real roots, $\pm c$ each with order two.

The four solutions to the velocity eigenvalue problem for the Dirac equation correspond to the the right and left handed electron and positron. That is, the velocity eigenstates of the Dirac equation are precisely the left and right-handed states used to represent fermions in the standard model.

There are two separate issues that might be causing downvotes (I didn't downvote yet, please fix). First, the Dirac Hamiltonian is in a discredited single-particle picture of the Dirac equation, where x is an operator describing the position of the electron. In the proper field theory picture, near Fock states have a momentum which is p and a velocity which is p/E in a wavepacket, and the two quantities can have simultaneous values (sort of, because particles are nonlocal). The other problem is that the equation you give for the speed eigenvalues has four solutions, (c,-c,ic,-ic).
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Ron MaimonDec 3 '11 at 6:01

As far as the problem with field theory versus QM goes, the velocity eigenstates of the electron are related to zitterbewegung (zbw) which has had a resurgence recently due to solid state physics research. So I'm not sure that it's discredited, for example, see the discussion of zbw and velocity eigenstates in Eur. Phys. J. B 83 , 301–317 (2011): arxiv.org/abs/1104.5632
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Carl BrannenDec 3 '11 at 8:03

I don't think it's completely discredited, it just needs a discussion--- the zbw is a property of positron states mixing with electron states in the single particle picture, its the electron zigging back and forth in time in the Feynman description. It's physical, but only in the Feynman form of particle dynamics, not so much in the field theory form. I am sure that this is the reason that a lot of people automatically downvote single particle discussions of Dirac eqn. I don't think it is nonsense, it contains a lot of physics, but it requires a careful discussion.
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Ron MaimonDec 3 '11 at 9:08

The argument that Heisenberg's Uncertainty Principle prohibits that we can know the exact values of momentum and velocity of a particle simultaneously is already discredited in the old textbook by Feynman on Quantum Electrodynamics.

Two observables can be simultaneously determined if the operators commute. For velocity and momentum, the operators commute $[\hat{p},\hat{v}]=0$; they do even in the Dirac wavefunction theory with its Zitterbewegung effects.